Let A, B and C be three points on the parabola y^2=6 x and let the line segment A B meet t…

Question

Let $A, B$ and $C$ be three points on the parabola $y^2=6 x$ and let the line segment $A B$ meet the line $L$ through $C$ parallel to the $x$-axis at the point $D$. Let $M$ and $N$ respectively be the feet of the perpend...

Answer: 36

Accepted Answer

Equation of $AB$

$y({t}_{1}+{t}_{2})=2x+2a{t}_{1}{t}_{2}$

$\begin{gathered} D,y=2a{t}_{3} \ \Rightarrow x=a({t}_{1}{t}_{3}+{t}_{2}{t}_{3}-{t}_{1}{t}_{2}) \ CD=|a({t}_{1}{t}_{3}+{t}_{2}{t}_{3}-{t}_{1}{t}_{3})-a{t}_{2}^{3}| \ AM=|2a{t}_{1}-2a{t}_{3}| \ BN=|2a{t}_{3}-2a{t}_{2}| \ {(\frac{AM\cdot BN}{CD})}^{2}={(\frac{4{a}^{2}({t}_{1}-{t}_{3})({t}_{3}-{t}_{2})}{a({t}_{1}{t}_{3}+{t}_{2}{t}_{3}-{t}_{1}{t}_{3}-{t}_{2}^{3})})}^{2} \ ={(\frac{4{a}^{2}({t}_{1}-{t}_{3})({t}_{3}-{t}_{2})}{a({t}_{1}-{t}_{3})({t}_{2}-{t}_{3})})}^{2} \ =16{a}^{2}=16\cdot {(\frac{3}{2})}^{2}=36 \end{gathered}$

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